THE LOCAL FORM OF DOUBLY STOCHASTIC MAPS AND JOINT ...

arXiv:math/0606060v1 [math.OA] 2 Jun 2006

THE LOCAL FORM OF DOUBLY STOCHASTIC MAPS AND JOINT MAJORIZATION IN II1 FACTORS MART´IN ARGERAMI AND PEDRO MASSEY Dedicated to our families Abstract. We find a description of the restriction of doubly stochastic maps to separable abelian C ∗ -subalgebras of a II1 factor M. We use this local form of doubly stochastic maps to develop a notion of joint majorization between n-tuples of mutually commuting self-adjoint operators that extends those of Kamei (for single self-adjoint operators) and Hiai (for single normal operators) in the II1 factor case. Several characterizations of this joint majorization are obtained. As a byproduct we prove that any separable abelian C ∗ -subalgebra of M can be embedded into a separable abelian C ∗ -subalgebra of M with diffuse spectral measure.

1. Introduction Majorization between self-adjoint operators in finite factors was introduced by Kamei [18] as an extension of Ando’s definition of majorization between self-adjoint matrices [4], a useful tool in matrix theory. Later on, Hiai considered majorization in semifinite factors between selfadjoint and normal operators [12, 13]. The reason why majorization has attracted the attention of many researchers (see the discussion in [13] and the references therein) is that it provides a rather subtle way to compare operators and occurs naturally in many contexts (for example [5, 10, 11]). Recently, majorization has regained interest because of its relation with norm-closed unitary orbits of self-adjoint operators and conditional expectations onto abelian subalgebras [5, 6, 8, 11, 15, 16, 20, 22]. One of the goals of this paper (section 4) is to obtain an extension of the notion of majorization between normal operators to that of joint majorization between n-tuples of commuting self-adjoint operators in a II1 factor (such extension is achieved in [19] for finite dimensional factors). In order to obtain characterizations of this extended notion we describe the local form of a doubly stochastic map (DS), i.e. we get a family of particularly well behaved DS maps that approximate the restriction of any DS map to separable abelian C ∗ -subalgebras of a II1 factor (section 3). As a byproduct, we construct separable abelian diffuse refinements of separable abelian C ∗ -subalgebras of a II1 factor M. This construction seems to have interest on its own. Some of the techniques we use seem to be new, even in the single element case. So far we have restricted our attention to the II1 factor case because, on one hand, technical aspects of the work become simpler and on the other hand, this is the context where majorization has its full meaning. Since every finite von Neumann algebra acting on a separable Hilbert space has a direct integral decomposition in terms of finite factors, the study of II1 factors provides useful information about more general algebras. The paper is organized as follows. In section 2 we recall some facts about abelian C ∗ subalgebras of a II1 factor. In section 3, after describing some technical results, we obtain a description of the local structure of doubly stochastic maps. In section 4 we introduce and develop the notion of joint majorization between finite abelian families of self-adjoint operators in a II1 factor and we obtain several characterizations of this relation. Finally, in section 6 we prove the results described in section 3. Supported in part by the Natural Sciences and Engineering Research Council of Canada. 2000 Mathematics Subject Classification: Primary 46L51; Secondary 46L10 . 1

2

MART´IN ARGERAMI AND PEDRO MASSEY

2. Preliminaries Throughout the paper M will be a II1 factor with normalized faithful normal trace τ . The C ∗ -subalgebras of M are always assumed unital. The subspace of self-adjoint elements of M will be denoted by Msa , and we will consider abelian families (a1 , . . . , an ) = (ai )ni=1 in Msa , that is finite families of mutually commuting self-adjoint operators in M. If (ai )ni=1 ⊆ Msa is an abelian family then C ∗ (a1 , . . . , an ) denotes the (unital) separable abelian C ∗ -subalgebra of M generated by the elements of the family. If A is an arbitrary abelian C ∗ -subalgebra of M then Γ(A) denotes its space of characters, i.e. the set of *-homomorphisms γ : A → C, endowed with the weak∗ -topology. The set Γ(A) is a compact space and A ≃ C(Γ(A)), where C(Γ(A)) denotes the C ∗ -algebra of continuous functions on Γ(A). 2.1. Joint spectral measures and joint spectral distributions. As we will consider a several-variable version of functional calculus, we state a few facts about it (see [23] for a different description). Let a ¯ =Q(ai )ni=1 be an abelian family in Msa . If A = C ∗ (a Q1n, . . . , an ), then Γ(A) n can be embedded in i=1 σ(ai ) ⊆ Rn . In fact, the map Φ : Γ(A) → i=1 σ(ai ) ⊆ Rn given by Φ(γ) = (γ(a1 ), . . . , γ(an )) is a continuous injection and therefore Γ(A) is homeomorphic to its image under this of the family and we Qn map; this image is called the joint spectrum a)) as C ∗ -algebras. If f ∈ C(σ(¯ a)), there denote it by σ(¯ a) ⊆ i=1 σ(ai ). Note that A ≃ C(σ(¯ exists a normal operator, denoted f (a1 , . . . , an ), that corresponds to f under the isomorphism A ≃ C(σ(¯ a)). This association extends the usual one variable functional calculus. If A ⊆ M is a separable C ∗ -subalgebra then Γ(A) is metrizable and the representation C(Γ(A)) ≃ A ⊆ M induces a spectral measure EA [9, IX.1.14] that takes values on the lattice P(M) of projections of M. Let µA be the (scalar) regular Borel measure on Γ(A) defined by µA (∆) = τ (EA (∆)). The regularity of µA follows from the fact Rthat every open set is σ-compact [21, 2.18]. The map Λ : L∞ (Γ(A), µA ) → M given by Λ(h) = Γ(A) h dEA is a normal ∗ -monomorphism (note that in this case the weak∗ topology of L∞ (Γ(A), µA ), restricted to the unit ball, is metrizable) and we have Z (1) τ (Λ(h)) = h dµA , ∀h ∈ L∞ (Γ(A), µA ). Γ(A)

We consider the von Neumann algebra L∞ (A) := Λ(L∞ (Γ(A), µA )) ⊆ M. When A = C ∗ (a1 , . . . , an ), Ea¯ := EA and µa¯ := µA are the joint spectral measure and joint spectral distribution of the abelian family a ¯ and we denote by Λa¯ : L∞ (Γ(¯ a), µa¯ ) → ∞ L (A) the normal isomorphism defined above. It is straightforward to verify that Λa¯ (πi ) = ai , 1 ≤ i ≤ n, and we write h(a1 , . . . , an ) := Λa¯ (h). In the case of a single self-adjoint operator a ∈ Msa the measure µa is the usual spectral distribution of a (see [8]), and it agrees with the Brown measure of a. In the particular case when x ∈ M is a normal operator, the real and imaginary parts of x are mutually commuting self-adjoint elements of M. Identifying the complex plane with R2 in the usual way, it is easy to see that the spectrum of x as a normal operator coincides with the joint spectrum of the abelian pair (Re(x), Im(x)), and that the spectral measure of x coincides with the joint spectral measure of (Re(x), Im(x)). ∼ 2.2. Comparison of measures and diffuse measures. We denote by M+ (Rn ) the Rset of all R n regular finite positive Borel measures ν on R with kζk dν(ζ) < ∞. We write ν(f ) = Rn f dν, ∼ for every ν ∈ M+ (Rn ) and every ν-integrable function f . In what follows, 1 denotes the constant function and πi : Rn → R denotes the projection onto the ith coordinate.

Definition 2.1. We say that µ is majorized by ν, and we write µ ≺ ν, if for Pmevery µ1 , . . . , µm ∈ Pm ∼ ∼ (Rn ) such that i=1 νi = ν, νi (1) = M+ (Rn ) with i=1 µi = µ there exist ν1 , . . . , νm ∈ M+ µi (1) and νi (πj ) = µi (πj ) for 1 ≤ i ≤ m, 1 ≤ j ≤ n.

THE LOCAL FORM OF DOUBLY STOCHASTIC MAPS AND JOINT MAJORIZATION IN II1 FACTORS 3

The relation ≺ in Definition 2.1 does not seem to be called “majorization” in the literature, ∼ but it will be a suitable name for us in the light of Theorem 4.5. If µ, ν ∈ M+ (Rn ) we shall write ν ∼ µ whenever ν(1) = µ(1) and ν(πj ) = µ(πj ) for every 1 ≤ j ≤ n. ∼ Theorem 2.2. [3, I.3.2] Let µ, ν ∈ M+ (Rn ). Then µ ≺ ν if and only if µ(f ) ≤ ν(f ) for every n continuous convex function f : R → R.

The next corollary is a direct consequence of Theorem 2.2 and the identity in equation (1). Corollary 2.3. Let a ¯ = (ai )ni=1 , ¯b = (bi )ni=1 ⊂ Msa be two abelian families. Then µa¯ ≺ µ¯b if and only if τ (f (a1 , . . . , an )) ≤ τ (f (b1 , . . . , bn )) for every continuous convex function f : Rn → R. We end this section with the following elementary fact about diffuse (scalar) measures, i.e. measures without atoms (recall that x is an atom of a measure µ if µ({x}) > 0). Lemma 2.4. Let K ⊂ Rn be compact and let µ be a regular diffuse Borel probability measure on K. Then for every α ∈ (0, 1) there exists a measurable set S ⊂ K such that µ(S) = α. 3. The local form of doubly stochastic maps A linear map Φ : M → M is said to be doubly stochastic [12] if it is unital, positive, and trace preserving. We denote the set of all doubly stochastic maps on M by DS(M). Doubly stochastic maps play an important role in the theory of majorization between selfadjoint operators (see for instance [1, 2, 12, 13]); thus, the study of their structure appears as a natural topic here. In what follows we introduce some terminology, we state Theorem 3.1, Proposition 3.2, and Lemma 3.5 and then we use them to prove Theorem 3.6. The proofs of these results will be presented at the end of the paper, in section 6. Although technical, they seem to have some interest on their own. Let A ⊆ M be an abelian C ∗ -subalgebra, and let EA and µA denote the spectral measure and the spectral distribution of A as defined in section 2.1. If x ∈ Γ(A) is such that EA ({x}) 6= 0, we say that x is an atom for EA . The set of atoms of EA is denoted At(EA ). Since µA = τ ◦EA , the faithfulness of the trace implies that At(µA ) =At(EA ). We say that A is diffuse if At(EA ) = ∅. The following theorem states that spectral measures of a separable A can be refined in a coherent way. Theorem 3.1. Let A ⊆ M be a separable abelian C ∗ -subalgebra. Then there exists a ∈ Msa such that C ∗ (A, a) is abelian and diffuse. Since the atoms of EA are in correspondence with the set of minimal projections of L∞ (A), Theorem 3.1 provides a way to embed A into a separable C ∗ -subalgebra A˜ = C ∗ (A, a) such ˜ has no minimal projections (see Remark 6.3 for further discussion). that L∞ (A) Proposition 3.2. Let B ⊂ M be a separable, diffuse, and abelian C ∗ -subalgebra. Then there exists an unbounded set M ⊆ N such that for every m ∈ M there exist k = k(m) partitions of t,m ′ the unity {qit,m }m ) = 1/m (1 ≤ i ≤ m, 1 ≤ t ≤ k), and i=1 ⊆ B ∩ M, 1 ≤ t ≤ k, with τ (qi t,m t,m such that for each b ∈ B, if we let βi = m τ (b qi ), then

! m k

1 X X t,m t,m

βi qi (2) lim b −

= 0. m→∞

k t=1

i=1

Remark 3.3. For fixed m and partitions of the unity {qit }m i=1 1 ≤ t ≤ k, the linear map ! m k 1X X b 7→ m τ (b qit ) qit k t=1 i=1

MART´IN ARGERAMI AND PEDRO MASSEY

4

is a contraction with respect to the operator norm. We denote by D(M) the convex semigroup D(M) = conv{Ad u : u ∈ U(M)}. 1 m Lemma 3.4. Let {pi }m i=1 , {qi }i=1 ⊆ M be partitions of the unity such that τ (pi ) = τ (qi ) = m , andPlet T ∈ DS(M). Then there exists ρ ∈ D(M) such that if β1 , . . . , βm ∈ R and αi = m m j=1 βj τ (T (qj ) pi ) for 1 ≤ i ≤ m, we have ! m m X X βi qi . αi pi = ρ (3) i=1

i=1

Proof. Let γi,j = m τ (T (qj ) pi ) ≥ 0; it is then straightforward to verify that (γi,j ) ∈ Rm×m is Pm a doubly stochastic matrix and, moreover, that αi = j=1 γi,j βj for every i = 1, . . . , m. By Birkhoff’s theorem the doubly stochastic P as a convex combination P matrix (γi,j ) can be written of permutation matrices, i.e. (γi,j ) = σ∈Sm ησ Pσ , where ησ ≥ 0, σ∈Sm ησ = 1 and Pσ is the m × m permutation matrix induced by σ ∈ Sm . Then we have

(4)

αi =

m X

γi,j βj =

j=1

X

ησ βσ(i)

1 ≤ i ≤ m.

σ∈Sm

The fact that M is a II1 factor and that the elements of the partitions {pi }i , {qi }i have the same trace guarantees the existence of unitaries uσ such that uσ qσ(i) (uσ )∗ = pi , 1 ≤ i ≤ m, for every σ ∈ Sm . Indeed, if σ ∈ Sm , the equalities, τ (qσ(i) ) = τ (pi ) imply that there exist ∗ ∗ partial isometries vi,σ ∈ M such that vi,σ vi,σ = pi and vi,σ vi,σ = qσ(i) for i = 1, . . . , m. Pm Then uσ = i=1 vi,σ ∈ M are the required unitaries. Using equation (4), and letting ρ(· ) = P ∗ σ∈Sm ησ uσ (· ) uσ , ! ! m m m X X X X X ∗ βσ(i) uσ qσ(i) uσ ησ ησ βσ(i) pi = αi pi = i=1

i=1

=

X

σ∈Sm

ησ uσ

i=1

σ∈Sm

σ∈Sm

m X i=1

βi qi

!

u∗σ

=ρ

m X i=1

βi qi

!

. 

′ Lemma 3.5. Let B ⊂ M be a separable C ∗ -subalgebra, and let {pi }m i=1 ⊆ B ∩ M be a partition of the unity. Then there exists a sequence {ρi }i∈N ⊂ D(M) such that for every b ∈ B, if we let βi (b) = τ (b pi )/τ (pi ), then

m

X

βi (b)pi = 0. lim ρj (b) − j→∞

i=1

Theorem 3.6. Let A, B ⊆ M be separable abelian C ∗ -subalgebras and let T ∈ DS(M). Let S be the operator subsystem of B given by S = T −1 (A) ∩ B. Then there exists a sequence (ρr )r∈N ⊆ D(M) such that limr→∞ kT (b) − ρr (b)k = 0 for every b ∈ S. Proof. First, note that we just have to prove the theorem for separable diffuse abelian C ∗ subalgebras of M; indeed, assume it holds for such algebras and let A, B ⊆ M be arbitrary separable abelian C ∗ -subalgebras. Then, by Theorem 3.1 there exist separable diffuse abelian ˜ Thus we get a sequence {ρr }r∈N ⊆ D subalgebras A˜ and B˜ of M such that A ⊆ A˜ and B ⊆ B. −1 ˜ ∩ B. ˜ So we assume such that limr→∞ kT (b) − ρr (b)k = 0, for every b ∈ T (A) ∩ B ⊆ T −1 (A) that A and B are diffuse. By Proposition 3.2, there exists an unbounded set M ⊆ N and, for each m ∈ M, k(m) j,m m ′ ′ partitions of the unity {qij,m }m i=1 ⊆ B ∩ M and {pi }i=1 ⊆ A ∩ M (in order to simplify

THE LOCAL FORM OF DOUBLY STOCHASTIC MAPS AND JOINT MAJORIZATION IN II1 FACTORS 5

the notation we avoid the supra-index m and write qij , pji ), 1 ≤ j ≤ k, such that for every b ∈ T (A)−1 ∩ B and every r ∈ N, there exists m0 (r, b) ∈ M such that if m ≥ m0 we have

!

m k X X

1

1 j j

βi qi < (5)

b − k

r

i=1 j=1 and

!

m k X X

1

j j

T (b) − 1 α p i i
0. Then there exists l ∈ N such that kb − bl k < ǫ/3. If r > max{l, 9/ǫ}, then kT (bl ) − ρr (bl )k < ǫ/3, and so kT (b) − ρr (b)k ≤ ǫ. 

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MART´IN ARGERAMI AND PEDRO MASSEY

Corollary 3.7. Let T ∈ DS(M) and let (ai )ni=1 , (bi )ni=1 ⊆ Msa be abelian families such that T (bi ) = ai for 1 ≤ i ≤ n. Then there exists a sequence (ρr )r∈N ⊆ D such that for 1 ≤ i ≤ n limr→∞ kai − ρr (bi )k = 0. Proof. Consider A = C ∗ (a1 , . . . , an ) and B = C ∗ (b1 , . . . , bn ), which are separable abelian C ∗ subalgebras of M. Applying Theorem 3.6 to this algebras we get a sequence (ρr )r∈N ⊆ D such that limr→∞ kT (b) − ρr (b)k = 0 for every b ∈ T −1 (A) ∩ B. By our choice, bi ∈ T −1 (A) ∩ B and r so kT (bi ) − ρr (bi )k = kai − ρr (bi )k − → 0.  4. Doubly stochastic kernels and joint majorization We begin by introducing doubly stochastic kernels, which are a natural generalization of doubly stochastic matrices. We shall use them to define joint majorization in analogy with [19]. Definition 4.1. Let (X, µX ), (Y, µY ) be two probability spaces. A positive unital linear map R ν : L∞ (Y, µY ) → L∞ (X, µX ) is said to be a doubly stochastic kernel if X ν(1∆ ) dµX = µY (∆), for every µY -measurable set ∆ ⊆ Y . Doubly stochastic kernels between probability spaces are norm continuous and normal. Example 4.2. Let X and Y be compact spaces and let µX and µY be regular Borel probability measures in X and Y respectively. Consider D ∈ L1 (µX × µY ) and let ν(f )(x) = R ∞ µX ) → L∞ (Y, µY ) is a doubly stochastic kernel if and Y D(x, y) f (y) dµY (y). Then ν : L (X, R R only if D(x, y) ≥ 0 (µX × µY )-a.e. and X D(x, y) dµX (x) = 1 µY -a.e, Y D(x, y) dµY (y) = 1 µX -a.e. In particular, if µX = µY is a measure with finite support {xi }m i=1 and such that 1 µX ({xi }) = m for 1 ≤ i ≤ m then D is a doubly stochastic kernel if and only if the matrix (D(xi , xj ))i, j is an m × m doubly stochastic matrix. Proposition 4.3. Let a ¯ = (ai )ni=1 , ¯b = (bi )ni=1 ⊆ Msa be abelian families. Then the following statements are equivalent: (1) There exists T ∈ DS(M) such that T (bi ) = ai , 1 ≤ i ≤ n. (2) There exists a doubly stochastic kernel ν : L∞ (σ(¯b), µ¯b ) → L∞ (σ(¯ a), µa¯ ) such that ν(πi ) = πi , 1 ≤ i ≤ n. Proof. Assume that T (bi ) = ai , 1 ≤ i ≤ n, with T ∈ DS(M). Let A = C ∗ (a1 , . . . , an ), B = C ∗ (b1 , . . . , bn ). As M is a finite von Neumann algebra, there exists a conditional expectation PA : M → L∞ (A) that commutes with τ . Then ν = Λa−1 ¯ ◦ PA ◦ T ◦ Λ¯ b is the desired doubly stochastic kernel. Conversely, let us assume the existence of ν as in 2. Let PB : M → L∞ (B) be the conditional expectation onto L∞ (B) that commutes with τ . Then define T = Λa¯ ◦ ν ◦ Λ¯b−1 ◦ PB ∈ DS(M). Clearly T (bi ) = ai , 1 ≤ i ≤ n.  Definition 4.4. Let a ¯ = (ai )ni=1 , ¯b = (bi )ni=1 be two abelian families in Msa . We say that a ¯ is jointly majorized by ¯b (and we write a ¯ ≺ ¯b) if there exists a doubly stochastic kernel ∞ ∞ ¯ ν : L (σ(b), µ¯b ) → L (σ(¯ a), µa¯ ) such that ν(πi ) = πi , 1 ≤ i ≤ n. If (x1 , . . . , xn ) is a finite family in M, let UM (x1 , . . . , xn ) denote the joint unitary orbit of the family with respect to the unitary group UM of M, i.e. UM (x1 , . . . , xn ) = {(u∗ x1 u, . . . , u∗ xn u) : u ∈ UM }. We shall also consider the convex hull of the unitary orbit of a family (xi )ni=1 , conv(UM (xi )ni=1 ) = {(ρ(xi ))ni=1 , ρ ∈ D}. We denote by conv(UM (xi )ni=1 ), convw (UM (xi )ni=1 ) and conv1 (UM (xi )ni=1 ) the respective closures in the coordinate-wise norm topology, coordinate-wise weak operator topology, and coordinatewise L1 topology.

THE LOCAL FORM OF DOUBLY STOCHASTIC MAPS AND JOINT MAJORIZATION IN II1 FACTORS 7

Theorem 4.5. Let a ¯ = (ai )ni=1 , ¯b = (bi )ni=1 be abelian families in Msa . Then the following statements are equivalent: (1) a ¯ is jointly majorized by ¯b. (2) a ¯ ∈ conv(UM (¯b)). (3) a ¯ ∈ conv 1 (UM (¯b)). (4) a ¯ ∈ conv w (UM (¯b)). (5) µa¯ ≺ µ¯b . (6) There exists a completely positive map T ∈ DS(M) such that ai = T (bi ), 1 ≤ i ≤ n. (7) There exists T ∈ DS(M) such that ai = T (bi ), 1 ≤ i ≤ n. (8) τ (f (a1 , . . . , an )) ≤ τ (f (b1 , . . . , bn )) for every continuous convex function f : Rn → R. Remark 4.6. Let x ∈ M be a normal operator. Recall (see the last paragraph of section 2.1) that there is a natural way to identify the usual spectral measure of x with that of the abelian pair (Re(x), Im(x)). If T ∈ DS(M), then since T is positive T (x) = y if and only if T (Re(x)) = Re(y) and T (Im(x)) = Im(y). From these facts and Theorem 4.5, we see that if x, y ∈ M are normal operators then x ≺ y in the sense of [13] if and only if (Re(x), Im(x)) ≺ (Re(y), Im(y)) in the sense of Definition 4.4. Let PN denote the trace preserving conditional expectation onto the abelian von Neumann subalgebra N ⊆ M. Using Theorem 4.5 we can then obtain a generalization of Theorem 7.2 in [8]. Corollary 4.7. Let N ⊆ M be an abelian von Neumann subalgebra and let (bi )ni=1 ⊆ Msa be an abelian family. Then (PN (bi ))ni=1 ≺ (bi )ni=1 . In the remainder of the section we prove the implications needed to prove Theorem 4.5. The single variable case of the following lemma can be found in [13]. Lemma 4.8. Let a ¯ = (ai )ni=1 , ¯b = (bi )ni=1 ⊆ Msa be abelian families. If a ¯ ∈ conv w (UM (¯b)) then there exists a completely positive T ∈ DS(M) such that ai = T (bi ), 1 ≤ i ≤ n. weakly

Proof. Let {(bj1 , . . . , bjn )}j∈J ⊆ conv(UM (b1 , . . . , bn )) such that bji −−−−→ ai , 1 ≤ i ≤ n. Then j

there exists a sequence (ρj )j∈J ⊆ D such that (bj1 , . . . , bjn ) = (ρj (b1 ), . . . , ρj (bn )), for every j ∈ J. Note that ρj is a completely positive doubly stochastic map and the net {ρj }j∈J is norm bounded. Therefore this net has an accumulation point in the BW topology [7], i.e. there exists a subnet (which we still call {ρj }j∈J ) and a completely positive map T : M → M such weakly

that ρj (x) −−−−→ T (x) if x ∈ M. By normality of the trace, T is trace preserving, positive and j

weakly

unital. Since ρj (bi ) = bji −−−−→ ai , we have T (bi ) = ai , 1 ≤ i ≤ n. j



Lemma 4.9. Let a ¯ = (ai )ni=1 , ¯b = (bi )ni=1 ⊆ Msa be abelian families. If a ¯ ≺ ¯b, then µa¯ ≺ µ¯b . ∞ Proof. By hypothesis, a ¯ ≺ ¯b; that is, there exists a doubly stochastic kernel ν : LP (σ(¯b), µ¯b ) → m ∞ ∼ n L (σ(¯ a), µa¯ ) such that ν(πi ) = πi , 1 ≤ i ≤ n. Let ν1 , . . . , νm ∈ M+ (R ) with j=1 νj = µa¯ . ′ ′ ′ Define measures ν j by νj (∆) = νj (ν(1∆ )). By continuity of ν, νj (f ) = νj (ν(f )) for every f ∈ L∞ (σ(¯b), µ¯b ). So ν ′j (πi ) = νj (ν(πi )) = νj (πi ), 1 ≤ i ≤ n and 1 ≤ j ≤ m, and similarly Pm Pm νj (1) = ν ′j (1), so that νj ∼ ν ′j , for 1 ≤ j ≤ m. Finally, j=1 ν ′j (∆) = j=1 νj (ν(1∆ )) = Pm µa¯ (ν(1∆ )) = µ¯b (∆). Therefore j=1 ν ′j = µ¯b . We conclude that µa¯ ≺ µ¯b . 

Lemma 4.10. Let a ¯ = (ai )ni=1 , ¯b = (bi )ni=1 ⊂ Msa be abelian families. If µa¯ ≺ µ¯b , then there exists T ∈ DS(M) such that T (bi ) = ai , 1 ≤ i ≤ n.

MART´IN ARGERAMI AND PEDRO MASSEY

8

m(r)

a) with diam(∆rj ) < 1/r Proof. By compactness, we can consider partitions {∆rj }j=1 of σ(¯ r r r r for every 1 ≤ j ≤ m. Fix points x1 , . . . , xm(r) with xj ∈ ∆j and define measures µrj by P µrj (· ) = µa¯ (· ∩ ∆rj ). Then clearly j µrj = µa¯ . As µa¯ ≺ µ¯b by hypothesis, there exist measures P νjr with νjr ∼ µrj and j νjr = µ¯b . Let gjr be the Radon-Nikodym derivatives gjr = dνjr /dµ¯b . P a) × σ(¯b) → R by Note that j gjr = 1 (µ¯b − a.e.). Define a function Dr : σ(¯ m(r)

Dr (s, t) =

X j=1

gjr (t) 1∆r (s). µa (∆rj ) j

We will use the kernels Dr to approximate T . Let us define νr : L∞ (σ(¯b), µ¯b ) → L∞ (σ(¯ a), µa¯ ) by Z b(t) Dr (s, t) dµ¯b (t). νr (b)(s) = σ(¯ b)

The map νr can be seen to be doubly stochastic using the equivalence µrj ∼ νjr . By Proposition R 4.3 there is an associated sequence {Tr }r ⊂ DS(M) such that Tr (bi ) = σ(¯a) νr (πi ) dEa¯ ∈ L∞ (A), 1 ≤ i ≤ n. The bounded net {Tr }r∈N has a subnet {Tk }k∈K that converges to a cluster point T ∈ DS(M) in the BW topology. Since this subnet is bounded, T (bi ) = w- limk∈K Tk (bi ) ∈ L∞ (A). We claim that T (bi ) = ai , 1 ≤ i ≤ n. To see this, since the net {Tk (bi )}k∈K is bounded, we just have to prove that lim τ (x Tk (bi )) = τ (x ai ), 1 ≤ i ≤ n, ∀x ∈ A. k

Equivalently, we have to show that for every continuous function f ∈ C(σ(¯ a)) and every i = 1, . . . , n, ! Z Z Z lim f (s) f (s) πi (s) dµa¯ (s). Dk (s, t) πi (t) dµ¯b (t) dµa¯ (s) = k

σ(¯ a)

σ(¯ b)

σ(¯ a)

This can be seen by a standard approximation argument, using the uniform continuity of f , the fact that the diameters of ∆rj tend to 0 as r increases, and the equivalence µrj ∼ νjr .  Proof of Theorem 4.5. Proposition 4.3 shows the equivalence (7)⇔(1) and Corollary 3.7 is (7)⇒(2). The implication (2)⇒ (3)⇒(4) is trivial. Lemma 4.8 shows that (4)⇒(6), and it is clear that (6)⇒(7). Lemmas 4.9, 4.10 and Proposition 4.3 prove the equivalence (5)⇔(1). So we have that (1)-(7) are equivalent. Finally, Corollary 2.3 shows that (5)⇔(8).  5. Joint unitary orbits of abelian families in Msa ¯ and ¯b are jointly approximately Given families a ¯ = (ai )ni=1 , ¯b = (bi )ni=1 ⊆ M, we say that a ¯ unitarily equivalent in M if a ¯ ∈ UM (b), that is if there exists a sequence of unitary operators (un )n∈N ⊆ M such that limn→∞ kun bi u∗n − ai k = 0 for every i = 1 . . . , n. It is clear that this is an equivalence relation. Moreover, if a ¯ and ¯b are jointly approximately unitarily equivalent in M then a ¯ is an abelian family if and only if ¯b is. In [8] a characterization of this equivalence relation between selfadjoint operators is obtained, in terms of the spectral distributions. The following results exhibits a list of characterizations of this relation for abelian families in Msa . Theorem 5.1. Let a ¯ = (ai )ni=1 and ¯b = (bi )ni=1 ⊂ Msa be abelian families. Then the following statements are equivalent: (1) a ¯ and ¯b are jointly approximately unitary equivalent in M. (2) a ¯ ≺ ¯b and ¯b ≺ a ¯ (3) µa¯ = µ¯b (4) τ (f (a1 , . . . , an )) = τ (f (b1 , . . . , bn )) for every continuous convex function f : Rn → R. (5) τ (f (a1 , . . . , an )) = τ (f (b1 , . . . , bn )) for every continuous function f : Rn → R.

THE LOCAL FORM OF DOUBLY STOCHASTIC MAPS AND JOINT MAJORIZATION IN II1 FACTORS 9

Proof. By Theorem 4.5 we have (1)⇒(2) and (2)⇔(4). Moreover, (4) is equivalent to µa¯ (f ) = µ¯b (f ) for every convex function f . Then µa¯ (f ) = µ¯b (f ) for every continuous function f [3, Proposition I.1.1], and this in turn implies that µa¯ = µ¯b . Therefore, (4)⇒(5)⇒(3). Again, by Theorem 4.5 (3)⇒(2) and so (2)-(5) are equivalent. Finally, we prove that (3)⇒(1). If we a) = supp µa¯ = supp µ¯b = σ(¯b) and for every Borel set ∆ in σ(¯ a) assume that µa¯ = µ¯b then, σ(¯ we have (13)

τ (Ea¯ (∆)) = µa¯ (1∆ ) = µ¯b (1∆ ) = τ (E¯b (∆)).

Let ǫ > 0. By compactness, choose B1 , . . . , Bm to be a finite disjoint covering of σ(¯ a) = σ(¯b) such that there are points xj ∈ Bj with the property that |πi (λ) − πi (xj )| < ǫ/2 for every λ ∈ Bj , 1 ≤ i ≤ n, 1 ≤ j ≤ m. Then we get, using the Spectral Theorem,





m m X X



ǫ ǫ

ai − πi (xj )E¯b (Bj ) πi (xj )Ea¯ (Bj ) < , bi −

< 2

2



j=1 j=1 for i = 1, . . . , n. From equation (13) we get that τ (Ea¯ (Bj )) = τ (E¯b (Bj )) for every j = 1, . . . , m. As in the proof of Lemma 3.4, we get a unitary wǫ ∈ U(M) such that wǫ∗ E¯b (Bj )wǫ = Ea¯ (Bj ) for every j. Then   m m X X πi (xj )Ea¯ (Bj ). πi (xj )E¯b (Bj ) wǫ = wǫ∗  j=1

j=1

Finally, for every i we have

 

m X

ǫ ∗ ∗

 πi (xj )E¯b (Bj ) wǫ k wǫ bi wǫ − ai k ≤ wǫ bi −

+ 2 < ǫ. 

j=1

Corollary 5.2. Let Θ be an automorphism of M. Then Θ|A is approximately inner for each separable abelian C ∗ subalgebra A ⊂ M. Proof. The uniqueness of the trace guarantees that Θ is trace-preserving. Being multiplicative, the range of an abelian set will be again abelian. So Θ is a DS map that takes an abelian family in M into another. Consider a countable dense subset {ai } of A, and use Theorem 5.1 to obtain unitaries un for each finite subset {a1 , . . . , an }. An ǫ/3 argument shows then that the sequence {Ad un } approximates Θ in all of A.  s

x) the closure in the coordinate-wise strong Given x¯ = (xi )ni=1 ⊆ M we denote by UM (¯ operator topology. An immediate consequence of Theorem 5.1 is that the norm closure of the unitary orbit of a selfadjoint abelian family in a II1 factor is strongly closed. This generalizes [8, Theorem 5.4] and [22, Theorem 8.12(1)]: Corollary 5.3. Let a ¯ = (ai )ni=1 ⊆ Msa be an abelian family. Then UM (¯ a)

kk

s

= UM (¯ a) .

s a) such that bji a) . There exists a net (bj1 , . . . , bjn )j∈J ⊆ UM (¯ Proof. Let ¯b = (bi )ni=1 ∈ UM (¯ n converges strongly to bi for each i = 1, . . . , n. Let f : R → R be a continuous function. Then τ (f (bj1 , . . . , bjn )) = τ (f (a1 , . . . , an )) for every j. Using [23, Lemma II.4.3] we conclude that τ (f (b1 , . . . , bn )) = τ (f (a1 , . . . , an )). So (5) of Theorem 5.1 implies that ¯b ∈ UM (¯ a). The other inclusion is trivial. 

6. Some technical results In this section we prove the results presented at the beginning of section 3. First, we show that any separable abelian C ∗ -subalgebra of M can be embedded into a separable diffuse abelian C ∗ -subalgebra. Then, we prove some approximation results that hold for separable diffuse abelian C ∗ subalgebras of M.

MART´IN ARGERAMI AND PEDRO MASSEY

10

6.1. Refinements of spectral measures. We begin by recalling some elementary facts about inclusions of abelian C ∗ algebras. If A ⊆ B are unital C ∗ -algebras, then the function Φ : Γ(B) → Γ(A) given by Φ(γ) = γ|A is a continuous surjection onto Γ(A). If we assume further that A ⊆ B ⊆ M are separable and that EA , EB denote their spectral measures, then EA = EB ◦Φ−1 and µA = µB ◦ Φ−1 . Note that At(µA ) =At(EA ) where P At(EA ) is the set of atoms of the spectral measure EA (see the beginning of section 3). Let x∈At(EA ) µA ({x}) be the total atomic mass of EA . Since µA is finite, the total atomic mass is bounded and thus, the set of atoms is countable set.

Lemma 6.1. With the notation above, if x ∈ At(EB ) then Φ(x) ∈ At(EA ), and the total atomic mass of B is smaller that the total atomic mass of A.

Proof. Let x ∈ At(EB ) and note that 0 6= µB ({x}) ≤ µB (Φ−1 (Φ({x}))) = µA (Φ({x})), so Φ(x) ∈ At(EA ) = At(µA ). We consider the equivalence relation in At(EB ) induced by Φ, i.e. x ∼ y if Φ(x) = Φ(y). If Q ∈ Q :=P At(EB )/ ∼ is such that Φ(x) = xQ for every x ∈ Q, then using that Q is countable we get x∈Q µB ({x}) = µB (Q) ≤ µB (Φ−1 ({xQ })) = µA ({xQ }). Therefore X X X X X µA ({x}). µB ({x}) = µB ({x}) ≤ µA (xQ ) ≤ x∈At(EB )

Q∈Q x∈Q

Q∈Q

x∈At(EA )

 Proposition 6.2. With the notations above, let x0 ∈ Γ(A) be an atom of EA and let α, β ∈ R with 0 < α < β. Then there exists a ∈ A′ ∩ Msa with [α, β] ⊆ σ(a) ⊆ [α, β] ∪ {0}, PR(a) = EA ({x0 }), and such that EB has no atoms in the fibre Φ−1 (x0 ), where B = C ∗ (A, a) ⊂ M. Proof. Let p = EA ({x0 }) and consider a masa A ⊂ M such that p ∈ A. Then pA is a masa 1 in the II1 factor pMp, where the trace is τp = τ (p) τ . It is well known that there exists a countably generated, non-atomic von Neumann subalgebra B of pA such that there is a von Neumann algebra isomorphism Φ : L∞ ([0, 1], m) → B, with m the Lebesgue measure on [0, 1], R1 and with τp (Φ(f )) = 0 f dm. Put a ˜ = Φ(id); it is clear that a ˜ has no atoms in its spectrum with the exception of 0, and that Ea˜ ({0}) = 1 − p, σ(a) = [0, 1]. Let a = (β − α)˜ a + α p, so [α, β] ⊆ σ(a) ⊆ [α, β] ∪ {0}, PR(a) = p = EA ({x0 }). As p is a minimal projection in L∞ (A), we have pb = pbp = λb p for every b ∈ A and so ab = apb = λb pa = bpa = ba. Thus a ∈ A′ ∩ M. Let B = C ∗ (A, a) and let Φ : Γ(B) → Γ(A), Ψ : Γ(B) → Γ(C ∗ (a)) be the continuous surjections induced by the inclusions A ⊆ B and C ∗ (a) ⊆ B. Note that the restriction Ψ|Φ−1 (x0 ) is injective. Indeed, let x, y ∈ Φ−1 (x0 ) be such that Ψ(x) = Ψ(y), i.e. the restriction of the characters to C ∗ (a) coincide. Since Φ(x) = Φ(y) (= x0 ), the characters also coincide on A and therefore are equal as characters in B, since B is generated by A and C ∗ (a). On the other hand, if x ∈ Γ(B) is such that x(a) 6= 0, then Φ(x) = x0 . Indeed, assume that Φ(x) 6= x0 . Let f ∈ C(Γ(A)) with f (Φ(x)) = 0 and f (x0 ) = 1. So f ◦ Φ ≥ 1Φ−1 (x0 ) . But then Z Z f ◦ Φ dEB ≥ 1Φ−1 (x0 ) dEB = EB (Φ−1 (x0 ) = EA ({x0 }) = p. Γ(B)

Γ(B)

Note that if 0 ∈ σ(a) then it is an isolated point, so in any case we have p ∈ C ∗ (a) ⊆ B. Then 0 = f ◦ Φ(x) ≥ x(p) ≥ 0, so x(p) = 0. Since 0 ≤ a ≤ β p, x(a) = 0 and the claim follows. Now let z ∈ Φ−1 (x0 ). If z(a) 6= 0, from the first part of the proof we deduce that Ψ−1 (Ψ(z)) = {z}. Therefore EB ({z}) = Ea ({Ψ(z})) = 0, since Ψ(z)(a) 6= 0 and At(Ea ) ⊆ {0}. If z(a) = 0, then {z} = Φ−1 (x0 ) \ {x ∈ Φ−1 (x0 ) : x(a) 6= 0} = Φ−1 (x0 ) \ Ψ−1 ({x ∈ Γ(C ∗ (a)) : x(a) 6= 0}) and EB (Ψ−1 ({x ∈ Γ(C ∗ (a)) : x(a) 6= 0})) = =

Ea ({x ∈ Γ(C ∗ (a)) : x(a) 6= 0}) EA ({x0 }) = EB (Φ−1 (x0 )).

THE LOCAL FORM OF DOUBLY STOCHASTIC MAPS AND JOINT MAJORIZATION IN II1 FACTORS11

From this we conclude that EB ({z}) = 0.



Proof of Theorem 3.1. Recall that the set At(EA ) of atoms of EA is a (possibly infinite) countable set. If At(EA ) = ∅ then EA is already diffuse and we are done. Otherwise, let us enumerate At(EA ) = {xi : 1 ≤ i ≤ r}, where r ∈ N ∪ {∞}. For 1 ≤ i ≤ r, let Ii = S S 1 1 , 1 + 2n−1 ]. Then Ii ∩ 1≤i6=j≤r Ij = ∅ and ri=1 Ii ⊆ [1, 2]. For each i = 1, . . . , r there [1 + 2n exists, by Proposition 6.2, ai ∈ A′ ∩Msa such that PR(ai ) = EA ({xi }), Ii ⊆ σ(ai ) ⊆ Ii ∪{0}, and

A denotes the continuous such that EAi has no atoms in the fibre Φ−1 i (xi ), where Φi : Γ(Ai ) →P r surjection induced by the inclusion A ⊆ Ai := C ∗ (A, ai ). Let a = i=1 ai ∈ A′ ∩ Msa (this sum converges because the ranges of the operators ai are orthogonal and kai k ≤ 2 for every i). Then B := C ∗ (A, a) is an abelian subalgebra of M. We claim that the spectral measure EB of B has no atoms. Indeed, first note that 1Ii ∈ C(∪1≤j≤r Ij ) is a continuous function (because the distance between the sets Ii and ∪i6=j Ij is positive); then, since 1Ii (a) = ai , it follows that Ai ⊂ B for every i = 1, . . . , r. Assume now that x ∈ At(Γ(B)) and let Φ : Γ(B) → Γ(A) be as before. By Lemma 6.1 there exists i ∈ {1, . . . , r} such that Φ(x) = xi ∈ At(EA ) . Since Φ = Φi ◦ Ψi , where Ψi : Γ(B) → Γ(Ai ) is the surjection induced by the inclusion Ai ⊆ B, we conclude that Ψi (x) ∈ Φ−1 i (xi ) is an atom of the measure EAi , again by Lemma 6.1. But this last assertion is a contradiction because by construction there are no atoms in the fibre Φ−1  i (xi ) by construction. Remark 6.3. Given an abelian C ∗ subalgebra A ⊂ M, a direct way to find an abelian C ∗ subalgebra A ⊆ A˜ ⊂ M with diffuse spectral measure is to consider a masa in M that contains A. The additional information we obtain from Theorem 3.1 is that A˜ can be chosen separable (as a C ∗ -algebra) whenever A is separable. When this is the case, the character space of A˜ is metrizable, a fact that is crucial for our calculations. 6.2. Discrete approximations in separable diffuse abelian algebras. Given a compact metric space it is always possible to find, using uniform continuity, discrete uniform approximations of continuous functions by linear combinations of characteristic functions of certain sets {Qi }m i=1 . But if we consider a measure on this space and we require equal measures for these sets, there might not be any good uniform approximation based on characteristic functions (even for measures of compact support in the real line). Proposition 3.2 is an intermediate solution to this problem. It was inspired by the proof of [13, Lemma 4.1].

Proof of Proposition 3.2. The space Γ(B) is a metrizable compact topological space, so we consider a metric d in Γ(B) inducing its topology. Let r ∈ N; by compactness, there exists P ˜ i ) < 1 and k0 µB (Q ˜ i }k0 of Γ(B) with diamd (Q ˜ i ) = 1. Let m = m(r) be a partition {Q i=1 i=1 r 2 ˜ such that 1/m ≤ min{µB (Qj ) : 1 ≤ j ≤ k0 }. Then for 1 ≤ j ≤ k0 there exists kj ∈ N ˜ ˜ j ) = kj /m + δj with 0 ≤ δj < 1/m. If we let k˜ = k(r) such that µB (Q = minj {kj } then −1 ˜ j ) − 1, 1 ≤ j ≤ k0 }. k˜ ≥ max{µB (Q ˜ with µB (Q ˜ t }kj of each Q ˜ j (1 ≤ t ≤ k), ˜ t ) = 1/m For t = 1, . . . , k0 , choose k˜ partitions {Q j,s s=0 j,s ˜ Note that we can ˜ t ) = δj , in such a way that Q ˜t ⊂ Q ˜ 1 , 2 ≤ t ≤ k. if 1 ≤ s ≤ kj and µB (Q j,0 j,0 j, t ˜t ⊆ Q ˜ 1 with µB (Q ˜ t ) = δj < 1/m, and always make such a choice: using Lemma 2.4 choose Q j,0 j, t j,0 k ˜ t } j of Q ˜j \ Q ˜ t using again Lemma 2.4 (note that µB (Q ˜j \ Q ˜t ) = then take a partition {Q j,s s=1 j,0 j,0 ′ t t ′ ˜ ∩Q ˜ = ∅ if t 6= t . kj /m). By this choice, Q j,0 j,0 P ˜ let Q ˜t ˜t ˜ t = ∪k0 Q For each t = 1, . . . , k, 0,0 j=1 j,0 . Then µB (Q0,0 ) = 1 − j kj /m = (m − P Pk0 t ˜ t n1 ˜ j kj subsets {Qi }i=1 of j=1 kj )/m. Finally, make partitions of each set Q0,0 into n1 = m − t t ˜ }i , we end up with k˜ partitions ˜ }j, s ∪ {Q measure 1/m. By re-labeling the k˜ partitions {Q i j,s t, m m ˜ {Qi }i=1 , for 1 ≤ t ≤ k, such that: ˜ 1. µB (Qt, m ) = 1/m, for every i ∈ {1, . . . , m}, t ∈ {1, . . . , k}; i

MART´IN ARGERAMI AND PEDRO MASSEY

12

m 2. diamd (Qt, ) ≤ 1/r, if i > n1 ; i ′ m ′ 3. if 1 ≤ i, i ≤ n1 then Qt, ∩ Qit ′ , m = ∅ if i 6= i ′ or t 6= t ′ . i m m Note that the construction of the k partitions {Qt, }i=1 was done in such a way that the i subsets that do not have small diameters are disjoint, even for different partitions. ˜ Let M = {m(r), r ≥ 1} and for every m = m(r) ∈ M let k(m) = k(r) as defined above and, m for i, t, m, let qit,m = EB (Qt, ). The set M is unbounded because the measure µB being diffuse i t,m m ˜ makes limr→∞ m(r) = ∞, and so limr→∞ k(r) = ∞. For each t = 1, . . . , k, {qi }i=1 ⊂ B ′ ∩ M is a partition of the unity. R Let b ∈ B, ǫ > 0, and let f ∈ C(Γ(B)) be such that b = Γ(B) f dEB . Then, by compactness, there exists δ > 0 such that if Q ⊆ Γ(B) with diamd (Q) < δ then diam(f (Q)) < ǫ. Let r ∈ N beRsuch that 1/r < δ and 2kbk/k(r) ≤ ǫ; let m = m(r) ∈ M, and let βit,m = m τ (b qit,m ) = m Qt,m f dµB . Properties 1-3 translate then into i

1’. τ (qit,m ) = 1/m, for every i ∈ {1, . . . , m}, t ∈ {1, . . . , k}; 2’. if i > n1 , then |f (x) − βit,m | ≤ ǫ, ∀x ∈ Qt,m i ; t,m t ′ ,m ′ 3’. if 1 ≤ i, i ≤ n1 then qi ⊥ qi ′ if i 6= i ′ or t 6= t ′ .

Therefore we have

! m m k k

1X

X 1 X X t,m t,m

t,m t,m βi qi βi qi = b−

b −

k

k t=1 i=1

t=1

i=1

k X m Z

1X

t,m = (b − βi ) dEB t,m

k

Qi t=1 i=1

n1 Z k X

1X

t,m ≤ (f − βi ) dEB + ǫ t,m

k

Qi t=1 i=1

n1 k X

2 kbk X 2kbk

t,m qi + ǫ = ≤ + ǫ ≤ 2ǫ

k k t=1 i=1

where the first inequality is a consequence of 2’ and the last equality follows from 3’.



Proof of Lemma 3.5. Fix a norm dense subset B = (bj )j∈N ⊆ B. In the construction leading to Dixmier’s Theorem, a previous result [17, 8.3.4] asserts that for each j, there exists a sequence n {ρnj }n∈N ⊆ D(M) such that for every 1 ≤ h ≤ j, kρnj (bh ) − τ (bh ) Ik − → 0. For each j ∈ N, let n0 = n0 (j) ∈ N be such that if n ≥ n0 then kρnj (bh ) − τ (bh ) Ik ≤ 1/j for 1 ≤ h ≤ j. If we let j

n (j)

→ 0 for every h ∈ N. Since (bj )j∈N is norm ρj = ρj 0 for j ∈ N, we get kρj (bh ) − τ (bh ) Ik − dense in B we have limj kρj (b) − τ (b) Ik = 0 for every b ∈ B. For every i = 1, . . . , m, consider the factor pi Mpi with (normalized) trace τi (pi x) = τ (xpi )/τ (pi ) . By the Dixmier approximation property mentioned in the first paragraph, applied to the separable C ∗ -subalgebra pi B of the finite factor pi Mpi , there exists a sequence {ρij }j∈N ∈ D(pi Mpi ) such that limj→∞ kρij (pi b) − τi (pi b)pi k = 0, for every b ∈ B. For each ρ ∈ D(pi Mpi ), we can consider an extension ρ˜ ∈ D(M) as follows: if ρ(pi b) = Pk P ∗ ˜ ∈Q D(M) by ρ˜(b) = kh=1 λh u ˜h b u ˜∗h , where h=1 λh uh b uh , with uh ∈ U(pi Mpi ), define ρ m i u ˜h = uh + (1 − pi ) ∈ U(M). If 1 ≤ i ≤ m set ρj = i=1 ρ˜j for j ≥ 1. It is easy to verify that if 1 ≤ i ≤ m then ρj (b pi ) = ρ˜ji (b pi ) for every b ∈ B. Then, if b ∈ B,



m m

X

X



βi (b)pi = ρ˜ji (b pi ) − τi (b pi )pi −−−→ 0. 

ρj (b) −

j→∞

i=1

i=1

THE LOCAL FORM OF DOUBLY STOCHASTIC MAPS AND JOINT MAJORIZATION IN II1 FACTORS13

Ackowledgements. We wish to thank Professors D. Farenick and D. Stojanoff for their support and useful discussions regarding the material in this paper. References [1] P.M. Alberti and A. Uhlmann, Stochasticity and partial order. Doubly stochastic maps and unitary mixing. Mathematische Monographien, 18. VEB Deutscher Verlag der Wissenschaften, Berlin, 1981. [2] P.M. Alberti, and A. Uhlmann, Dissipative motion in state spaces. Teubner-Texte zur Mathematik, 33. BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1981. [3] E.M. Alfsen,Compact Convex Set and Boundary Integrals, Springer-Verlag, New York, NY 1971. [4] T. Ando, Majorization, doubly stochastic matrices and comparison of eigen- values, Lecture Notes, Hokkaido Univ., 1982. [5] J. Antezana, P. Massey, and D. Stojanoff, Jensen’s Inequality and Majorization, preprint http://xxx.lanl.gov/abs/math.FA/0411442. [6] J. Antezana, P. Massey, M. Ruiz and D. Stojanoff, The Schur-Horn theorem for operators and frames with prescribed norms and frame operator, to appear in Illinois J. of Math. http://xxx.lanl.gov/abs/math.FA/0508646 [7] W. Arveson, Subalgebras of C ∗ -algebras Acta Math. 123 (1969), 141–224. [8] W. Arveson and R. Kadison, Diagonals of self-adjoint operators, preprint http://xxx.lanl.gov/abs/math.OA/0508482. [9] J.B. Conway, A course in functional analysis, Springer-Verlag, New York, NY 1990. [10] T. Fack, Sur la notion de valeur caract´ eristique, J. Operator Theory (1982), 307-333. [11] D.R. Farenick and S.M. Manjegani, Young’s Inequality in Operator Algebras, J. Ramanujan Math. Soc. 20 (2005), no. 2, 107–124. http://xxx.lanl.gov/abs/math.OA/0303318. [12] F. Hiai, Majorization and Stochastic maps in von Neumann algebras, J. Math. Anal. Appl. 127 (1987), no. 1, 18–48. [13] F. Hiai, Spectral majorization between normal operators in von Neumann algebras, Operator algebras and operator theory (Craiova, 1989), 78–115, Pitman Res. Notes Math. Ser., 271, Longman Sci. Tech., Harlow, 1992. [14] F. Hiai, Y. Nakamura, Closed Convex Hulls of Unitary Orbits in von Neumann Algebras, Trans. Amer. Math. Soc. 323 (1991), 1-38. [15] R. Kadison, The Pythagorean theorem I: the finite case, Proc. N.A.S. (USA), 99(7):4178-4184, 2002. [16] R. Kadison, The Pythagorean theorem II: the infinite discrete case, Proc. N.A.S. (USA), 99(8):5217-5222, 2002. [17] Kadison and Ringrose, Fundamentals of the Theory of Operator Algebras, Vol II, Academic Press, Orlando, Florida, 1986. [18] E. Kamei,Majorization in finite factors, Math. Japonica 28, No. 4 (1983), 495-499. [19] F.D. Mart´ınez Per´ıa, P. Massey, L. Silvestre, Weak matrix majorization. Linear Algebra Appl. 403 (2005), 343–368. [20] A. Neumann, An infinite-dimensional version of the Schur-Horn convexity theorem, J. Funct. Anal. 161 (1999), 418-451. [21] W. Rudin, Real and complex analysis. Third edition. McGraw-Hill Book Co., New York, 1987. [22] D. Sherman, Unitary orbits of normal operators in von Neumann algebras, Journal f¨ ur die reine und angewandte Mathematik, to appear. [23] M. Takesaki, Theory of Operator Algebras I, Encyclopaedia of Mathematical Sciences V, Springer Verlag, 2nd printing of the First Edition 1979. Department of Mathematics and Statistics, University of Regina, Saskatchewan,Canada S4S 0A2, [email protected] ´ tica, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Departamento de Matema Argentina, [email protected]